提出统一框架MAPCA，实现尺度不变表示学习，可灵活控制特征压缩方向。
Metric-Aware Principal Component Analysis (MAPCA):A Unified Framework for Scale-Invariant Representation Learning
- 基于广义特征值问题构建，通过度量矩阵M调节表示几何结构。
- β=0时为标准PCA，β=1时为输出白化，中间值实现连续谱偏置控制。
- 揭示输入/输出白化本质差异，统一Barlow Twins等自监督方法的几何意义。
我们提出度量感知主成分分析（MAPCA），一种基于广义特征值问题 max Tr(W^T Sigma W) 且满足 W^T M W = I 的统一框架，其中 M 为对称正定度量矩阵。M 的选择决定表示几何。规范的 beta 家族 M(beta) = Sigma^beta（β ∈ [0,1]）在标准 PCA（β=0）与输出白化（β=1）之间提供连续谱偏置控制，条件数 kappa(beta) = (λ₁/λ_p)^(1−β) 单调递减至各向同性。对角度量 M = D = diag(Sigma) 恢复了基于 Frisch (1928) 对角回归的不变主成分分析（IPCA），是该框架的特例。我们证明：尺度不变性成立当且仅当度量在缩放下满足变换规则 M_tilde = CMC，该条件仅精确满足于 IPCA，而不适用于中间 β 值的通用 beta 家族。除经典解释外，MAPCA 提供统一的几何语言，整合多种自监督学习目标：Barlow Twins 和 ZCA 白化对应 β=1（输出白化）；VICReg 的方差项对应对角度量。关键发现：尽管 W-MSE 被描述为基于白化的算法，其实际对应于 M = Sigma^{-1}（β = -1），完全超出谱压缩范围，且谱方向与 Barlow Twins 相反。这一输入/输出白化的本质区别在损失函数层面无法察觉，仅在 MAPCA 框架中清晰显现。
原文摘要 · Abstract (English)
We introduce Metric-Aware Principal Component Analysis (MAPCA), a unified framework for scale-invariant representation learning based on the generalised eigenproblem max Tr(W^T Sigma W) subject to W^T M W = I, where M is a symmetric positive definite metric matrix. The choice of M determines the representation geometry. The canonical beta-family M(beta) = Sigma^beta, beta in [0,1], provides continuous spectral bias control between standard PCA (beta=0) and output whitening (beta=1), with condition number kappa(beta) = (lambda_1/lambda_p)^(1-beta) decreasing monotonically to isotropy. The diagonal metric M = D = diag(Sigma) recovers Invariant PCA (IPCA), a method rooted in Frisch (1928) diagonal regression, as a distinct member of the broader framework. We prove that scale invariance holds if and only if the metric transforms as M_tilde = CMC under rescaling C, a condition satisfied exactly by IPCA but not by the general beta-family at intermediate values. Beyond its classical interpretation, MAPCA provides a geometric language that unifies several self-supervised learning objectives. Barlow Twins and ZCA whitening correspond to beta=1 (output whitening); VICReg's variance term corresponds to the diagonal metric. A key finding is that W-MSE, despite being described as a whitening-based method, corresponds to M = Sigma^{-1} (beta = -1), outside the spectral compression range entirely and in the opposite spectral direction to Barlow Twins. This distinction between input and output whitening is invisible at the level of loss functions and becomes precise only within the MAPCA framework.
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